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Probabilistic Semantics for Modal Logic.

Lando, Tamar Ariela.

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Probabilistic Semantics for Modal Logic.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Probabilistic Semantics for Modal Logic./
Author:	Lando, Tamar Ariela.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2012,
Description:	137 p.
Notes:	Source: Dissertations Abstracts International, Volume: 74-04, Section: A.
Contained By:	Dissertations Abstracts International74-04A.
Subject:	Logic. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3527187
ISBN:	9781267613073

Probabilistic Semantics for Modal Logic.
Lando, Tamar Ariela.

Probabilistic Semantics for Modal Logic. - Ann Arbor : ProQuest Dissertations & Theses, 2012 - 137 p.

Source: Dissertations Abstracts International, Volume: 74-04, Section: A.

Thesis (Ph.D.)--University of California, Berkeley, 2012.

This item must not be sold to any third party vendors.

We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940's. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero . In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfied in a model over the algebra if it is assigned to the top element in the algebra-or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propositional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4 C, is sound and complete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators). The connection with Tarski's topological semantics is developed throughout the text, and the first substantive chapter is devoted to a new and simplified proof of Tarski's completeness result via well-known fractal curves. This work may be applied in the many formal areas of philosophy that exploit probability theory for philosophical purposes. One interesting application in metaphysics, or mereology, is developed in the introductory chapter. We argue, against orthodoxy, that on a 'gunky' conception of space-a conception of space according to which each region of space has a proper subregion-we can still introduce many of the usual topological notions that we have for ordinary, 'pointy' space.

ISBN: 9781267613073Subjects--Topical Terms:

529544
Logic.
Subjects--Index Terms:

Completeness

Probabilistic Semantics for Modal Logic.
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100 1 $a Lando, Tamar Ariela. $3 3562966
245 1 0 $a Probabilistic Semantics for Modal Logic.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2012
300 $a 137 p.
500 $a Source: Dissertations Abstracts International, Volume: 74-04, Section: A.
500 $a Publisher info.: Dissertation/Thesis.
500 $a Advisor: Mancosu, Paolo;Stroud, Barry.
502 $a Thesis (Ph.D.)--University of California, Berkeley, 2012.
506 $a This item must not be sold to any third party vendors.
520 $a We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940's. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero . In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfied in a model over the algebra if it is assigned to the top element in the algebra-or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propositional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4 C, is sound and complete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators). The connection with Tarski's topological semantics is developed throughout the text, and the first substantive chapter is devoted to a new and simplified proof of Tarski's completeness result via well-known fractal curves. This work may be applied in the many formal areas of philosophy that exploit probability theory for philosophical purposes. One interesting application in metaphysics, or mereology, is developed in the introductory chapter. We argue, against orthodoxy, that on a 'gunky' conception of space-a conception of space according to which each region of space has a proper subregion-we can still introduce many of the usual topological notions that we have for ordinary, 'pointy' space.
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650 4 $a Logic. $3 529544
650 4 $a Mathematics. $3 515831
650 4 $a Philosophy. $3 516511
653 $a Completeness
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653 $a Probabilistic semantics
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690 $a 0422
710 2 $a University of California, Berkeley. $b Philosophy. $3 1683966
773 0 $t Dissertations Abstracts International $g 74-04A.
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